ABSTRACT

BORDEN, MARGARET LEAK. A Framework of Core Teaching Practices for Implementing Project-Based Learning (PBL) in the Mathematics Classroom (Under the direction of Dr. Erin Krupa).

Project-based learning (PBL) is a teaching methodology that utilizes projects to

engage students in learning the content through authentic experiences. PBL has been shown

to successfully motivate students to learn as well as reinvigorate teachers, and it can be used

to teach any discipline. However, it can be challenging to teach using PBL, and particularly

difficult for math teachers who tend to face more challenges than other disciplines when

teaching utilizing PBL. In order to make PBL more accessible to math teachers, this paper

merges several existing PBL frameworks with the various recommendations for math

education into one framework of core teaching practices for implementing PBL in a math

classroom. The framework describes what teachers should think about while planning the

project, what they need to be doing while students are working on the project, and how to

best support students when they are ready to share their work with others. The paper includes

multiple examples to help teachers visualize running a PBL in their classroom, and goes into

detail with one example to explain the different aspects of the framework. After reading this

paper, math teachers will feel empowered to be able to implement PBL in their classrooms,

non-classroom educators will be able to support their math teachers in implementing PBL,

and the myth that PBL cannot be used in math will be expelled.

© Copyright 2021 by Margaret Borden

All Rights Reserved

A Framework of Core Teaching Practices for Implementing Project-Based Learning (PBL) in the Mathematics Classroom

by Margaret Leak Borden

A thesis submitted to the Graduate Faculty of North Carolina State University

in partial fulfillment of the requirements for the degree of

Master’s of Science

Mathematics Education

Raleigh, North Carolina 2021

APPROVED BY:

_______________________________ _______________________________ Dr. Erin Krupa Dr. Cyndi Edgington Committee Chair _______________________________ _______________________________ Dr. Karen Hollebrands Dr. Kevin Gross

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DEDICATION

I would like to dedicate this paper to my daughter, Ellamarie Councill Borden,

because I want her to see that it is possible to pursue your dreams and make them your

reality, even if it takes a lot of hard work.

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BIOGRAPHY

Margaret Borden was born and raised in Winston-Salem, North Carolina. She moved

to Raleigh when she was accepted at NC State University in the undergraduate math

education program. She was awarded the Park Scholarship, a program which focuses on

scholarship, leadership, character, and service, and the experiences she had with her cohort

through that program inspired her to study abroad, triple major, and take on many different

leadership roles. Because she added a math degree and a communication degree to her

studies in math education, she needed to stay at NC State for a few more semesters past her

Park Scholarship allotment. She was awarded the Noyce Scholarship, a program that

supports STEM educators in their last few years in college who are interested in teaching in

underserved schools. Due to these two programs, Margaret was mentored by amazing math

education professors and math educators, who helped her to find presentation opportunities

and share her ideas with people who share her passions. She presented at various math

education conferences and led the college’s chapter of NCCTM, lighting a fire in her for

math education that extended beyond her coursework. It was through these experiences that

she discovered project-based learning (PBL), and she spent as much time as she could

learning about PBL, attending trainings on how to teach with PBL, and even presenting at the

NCCTM conference on it.

After graduating with her three degrees and a minor in Spanish, Margaret taught at

Knightdale High School of Collaborative Design (KHSCD). KHSCD was known for

utilizing innovative practices to find ways to motivate learners who did not respond well to

traditional schooling methods. They were well-versed in PBL, so it was a great fit for

Margaret since she wanted the opportunity to teach in an environment that supported PBL.

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She taught there for three and a half years, and found that it was exhilarating to work

alongside innovative educators, but it was also frustrating to do this work within the confines

of traditional grading practices, schedules, and other constraints.

While at KHSCD, Margaret married her husband, Peter, who was in law school at the

time. Simultaneously, Margaret decided to go to graduate school to pursue a Master’s in

Math Education, and worked on that part time while teaching full time. Peter now works as a

civil litigator in downtown Raleigh, and they moved to a home near Lake Wheeler to be

closer to his family. They decided to start their family, and learned from one of Margaret’s

mentors that there was a possibility to be funded if she finished her Master’s as a full-time

student. Thus, Margaret spent her last year at KHSCD pregnant with their daughter, who was

born in the beginning of the global Covid-19 pandemic, and then transitioned out of the

classroom the following fall to finish her Master’s as a graduate research assistant.

She spent the last year of her Master’s working on a PBI Global team with Dr. Hiller

Spires and Dr. Erin Krupa, which gave her the opportunity to work with interdisciplinary

projects at two different schools. She was able to serve as math support to the math teachers

on the team, helping them understand how their standards could be adequately addressed in

relation to the project context. She also took on a coaching role with math educators at a

charter school that is attempting to become a PBL school, helping them learn how to create

the classroom environment that could be conducive to effective PBL. All of these

experiences helped to inspire this thesis. Margaret will be pursuing her Ph.D. in Math

Education starting in the fall, and will continue working with Dr. Krupa on a different PBL

team that is specifically oriented towards high school math teachers. Margaret is very excited

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to continue her work and looks forward to the opportunity to follow some of her own

recommendations during her doctoral pursuits.

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ACKNOWLEDGMENTS

I would like to acknowledge my advisor Dr. Erin Krupa for all of her support

throughout this process. Dr. Krupa received an email with my resume and saw potential for

this partnership, and I am incredibly grateful that she decided to give me this opportunity.

She has believed in my ideas every step of the way, always seeking clarification in a way that

made them stronger. I am so excited to continue working with her throughout my doctoral

studies and career.

I would also like to acknowledge the other two math education representatives on my

committee: Dr. Cyndi Edgington and Dr. Karen Hollebrands. They have both been my

mentors throughout my career at NC State, presenting alongside me, thinking of me for

opportunities, and spending time advising me. They are two of my greatest cheerleaders and I

know that I can always reach out to them when I need support, resources, or a

recommendation. I look forward to more opportunities to work with them during my doctoral

studies and through my career.

Additionally, I would like to acknowledge the statistics representative on my

committee: Dr. Kevin Gross. Dr. Gross speaks about statistics in the same way that I feel

about statistics, with enthusiasm and curiosity. It stood out to me that he took the time in

class to explain the etymology of any new terminology, sometimes to help us remember the

term better and other times to help us understand the concept around it. I am so thankful that

he was willing to serve on my committee and provide a different perspective to the feedback.

There are many other people who have been a part of my journey as a math educator

who deserve to be recognized as well. Allison McCulloch has been one of my mentors from

the beginning and has always encouraged me to find opportunities to present as well as

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investigate my ideas further. Teresa Pierrie has been instrumental in developing me as an

educator and sending opportunities my way to support that development. Matthew Campbell

has mentored me through the process of going back to school and provided important

insights into how to continue to be connected to the classroom even when outside of the

classroom. My teammates at KHSCD challenged my ideas and pushed me to see many

different angles in each pedagogical approach, especially Tracy Pratt and Laura Harrell who

regularly spent hours in conversation with me about different math education ideas.

Similarly, I have been inspired by conversations I’ve had with my students, mentees and my

curriculum writing team, and continually seek to learn how everyone experiences these ideas

from their perspectives. Most notably, I have appreciated my student, Kiara Bush, for her

efforts on my projects and continued work with me to share the word about PBL in math and

the joys of math in general. Additionally, I have benefitted tremendously with my work with

math teachers from Wake STEM, PECIL, and CLA who are all at different points in the

journey to teaching with PBL, and have enjoyed learning from the PBI Global team.

Finally, I would like to acknowledge my family’s support. My husband, Peter

Borden, has always been so supportive of me, encouraging me to work hard and when my

hard work still fell short. He has taken on more work around the house and with the baby

when I’ve had to work outside of daycare hours in order to meet deadlines, and listened to

me as I talked way too much about my thesis and my classes. He is my rock and my biggest

cheerleader and I am so thankful to have him by my side throughout this experience. My

parents, in-laws, grandparents, sister, and friends have also all played a huge role in helping

me get to this point. They have picked up the phone to listen to my worries, taken me to

dinners to celebrate when I was awarded funding, and tried to understand what I’m talking

viii

about when I start passionately talking about math education. I feel so blessed to have so

many people who help me believe that I am capable of doing this work and being successful

when pursuing my passion.

I have really appreciated the incredible support system that has surrounded me as I

learn and grow as a math educator. I love to learn new things and meet new people, and feel

thankful to the people that I have met through NC State and Wake County that have helped

to make that learning possible.

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TABLE OF CONTENTS

LIST OF TABLES ................................................................................................................ x LIST OF FIGURES.............................................................................................................. xi Chapter 1: Introduction ...................................................................................................... 1 Brief History of PBL and Mathematics .................................................................................. 1 Statement of the Problem ...................................................................................................... 4 Chapter 2: Literature Review ............................................................................................. 6 Motivation for Using PBL in the Mathematics Classroom ..................................................... 6 Barriers to PBL in the Mathematics Classroom ..................................................................... 9 Effective Mathematics Teaching Practices........................................................................... 12 Effective Project-Based Teaching Practices ......................................................................... 15 Using PBL in a Mathematics Classroom ............................................................................. 19 Chapter 3: Discussion of Theoretical Framework ........................................................... 26 Connecting the Effective Teaching Practices ....................................................................... 26

Core Practice 1: Disciplinary ................................................................................... 26 Core Practice 2: Authenticity ................................................................................... 32 Core Practice 3: Iterative ......................................................................................... 36 Core Practice 4: Collaboration ................................................................................. 40

Framework for PBL in Mathematics Classroom .................................................................. 41 Planning Phase ........................................................................................................ 42

Align to Standards ........................................................................................ 42 Working Phase ........................................................................................................ 43

Engage in Disciplinary Practices .................................................................. 43 Build Procedural Fluency ............................................................................. 47 Assess Student Progress ............................................................................... 48

Sharing Phase .......................................................................................................... 50 Find Authentic Audience .............................................................................. 50

Chapter 4: Conclusion ...................................................................................................... 52 Limitations and Recommendations ...................................................................................... 52 Conclusion .......................................................................................................................... 53

REFERENCES ................................................................................................................... 55 APPENDIX......................................................................................................................... 60

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LIST OF TABLES

Table 3.1 Smith and Stein’s (2011) Five Practices for Orchestrating Productive Mathematical Discussions ................................................................................ 29

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LIST OF FIGURES

Figure 2.1 Teaching and Learning Practices. ..................................................................... 12

Figure 2.2 PBLWorks Gold Standard Design Elements (left) and Teaching Practices (right). ................................................................................ 15 Figure 2.3 The Core Practices of Project-Based Teaching.................................................. 16

Figure 3.1 Grossman, et. al.’s (2019) Core Practices of Project-Based Teaching Framework ........................................................................................ 27

Figure 3.2 Gold Standard PBL: Design Elements (left) and Teaching Practices (right). ................................................................................ 27

Figure 3.3 NCTM Effective Mathematics Teaching Practices ............................................ 28

Figure 3.4 Teaching Practices for PBL in the Mathematics Classroom .............................. 42

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CHAPTER 1: INTRODUCTION

It is not uncommon to hear someone say “I am not a math person,” or “I was never

very good at math.” It is a pervasive attitude shared amongst students, parents, non-math

teachers, and administrators. Even some math teachers believe that some people just aren’t

good at math. Boaler (2016) examined these fixed mathematical mindsets and learned that

these beliefs were damaging to a student’s ability to learn. However, there is hope! Her

research found that students can develop a growth mindset in mathematics, especially in a

math classroom where “mathematics is taught as a creative and open subject, all about

connections, learning, and growth, and [where] mistakes are encouraged” (Boaler, 2016,

p.20). Project-based learning (PBL) is a teaching methodology that is designed to foster this

type of learning. Although there are not a lot of examples of how PBL has been used in the

mathematics classroom, there is a large body of research on effective teaching practices in

both PBL classrooms and math classrooms. Thus, this paper will examine those practices,

outline the connections that can be drawn between them, and then recommend a framework

for utilizing PBL to support mathematical learning. To begin, the following two sections will

discuss the historical developments in education related to PBL and mathematics, and then

describe the problem still facing the effort to implement PBL in the math classroom.

Brief History of PBL and Mathematics

PBL is a relatively new term but, according to Knoll’s (1997) historical compilation

of the use of projects as a teaching methodology, the concept dates back centuries. It has its

origins in architecture and engineering education, where it was realized that learning by

lecture was not enough to become prepared to be successful as an architect or engineer

(Knoll, 1997). Just because students understand how a building should stand up on its own

2

does not mean that they are capable of building a structure that actually can stand up on its

own. Projects have since become the foundation of many other vocational disciplines, such

as medical education, that recognize the act of doing a job well cannot solely be learned in

books and lectures (Knoll, 1997). As Condliffe, et. al. (2016) points out in their literature

review of PBL, many school reforms have been based on the goal of graduating students who

are college and career ready. It stands to reason that utilizing the educational strategies that

careers employ, such as PBL, would help schools reach the goal of career readiness.

It is also not a new idea to use PBL in the K-12 learning environment. According to

Condliffe, et. al. (2016), PBL became popular to use in public education as part of the

progressive movement, coupled with the constructivist philosophy. Around the same time,

according to Klein’s (2003) history on mathematics education, there was a great debate

around which mathematical ideas are important to teach and how to teach them. The

progressive movement had sparked educational reforms in many subjects, and math teachers

were struggling to figure out how to utilize constructivist pedagogies, such as PBL, in their

practice (Condliffe, et. al., 2016). In response, the National Council for Teachers of

Mathematics (NCTM) advocated for mathematics students to spend more time problem-

solving and utilizing appropriate technology, and less time practicing tedious algorithms

(Klein, 2003; Confrey & Krupa, 2011). This period of education reform, from Klein’s (2003)

point of view, sparked a struggle between math education activists and parents, all trying to

decide if math education should be more geared towards career and citizen experiences

beyond school years or focus on preparing for academic rigor. This struggle still continues

today, but NCTM stated then and continues to advocate that mathematics education should

evolve with the needs of the current times, not rely on past ways that served needs that no

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longer exist (NCTM, 2018). PBL, as described by PBLWorks (2019), an organization that

seeks to support educators in designing quality PBL experiences for their students, provides

structures that can support students as they solve problems and apply mathematical

procedures and concepts in an academically rigorous way. In their book on how to

implement PBL in the math classroom, Fancher and Norfar (2019) assure teachers that PBL

done well will not adversely affect academic rigor and performance on standardized tests, but

rather support deeper learning that allows students to retain more information over time.

According to Confrey & Krupa’s (2011) account of the development of the Common

Core State Standards (CCSS), a lot of work has been done over the past fifty years by a

variety of experts to support math teachers in the development of quality mathematical

curricula. There have been many iterations of content standards and process standards, with

the CCSS being the most comprehensive and widely used set to be developed (Confrey &

Krupa, 2011). The focus in CCSS on students developing conceptual understanding of each

idea aligns well with the goals of PBL, Condliffe, et. al. (2016) points out. Thus, it is

reasonable to believe that PBL would be an effective pedagogical tool for a mathematics

classroom. Additionally, studies, such as Capraro and Capraro’s (2015) longitudinal study of

the effects of PBL in secondary STEM classrooms on student learning, have shown that the

use of PBL has a positive effect on student growth, especially that of low achievers.

Condliffe, et. al. (2016) adds that many studies have shown PBL to have “positive effects on

students’ engagement, motivation, and beliefs in their own efficacy” (p. iii). Although they

go on to explain that very few empirical studies have focused on math classrooms, the ones

that did found that students who studied math in a PBL setting outperformed their

counterparts learning math in a traditional setting (Condliffe, et. al., 2016). Therefore, using

4

PBL to teach math has the potential to positively affect students from all learning

backgrounds and in all desired ways.

Statement of the Problem

Despite the many similarities between the recommended practices from PBL

advocates and mathematics education advocates, math teachers have struggled to incorporate

PBL into their instruction (Condliffe, et. al., 2016). There have even been instances of

administrators implementing schoolwide initiatives to teach using PBL, but allowing their

math teachers to choose whether or not they utilize it (S. Gibbons, personal communication,

March 12, 2021). Although much has been written on the implementation of PBL in general

classrooms, very few of the examples given are for mathematics classrooms, and of those

few examples, most are for elementary and middle school math classes (Condliffe, et. al.,

2016; Fancher & Norfar, 2019). Aslan and Reigeluth (2015) examined the challenges related

to learner-centered education, and explained that much of this difficulty comes from teachers

who do not believe that PBL can help them address deficits in prior mathematical knowledge

and meet state standards in the time they are given to teach their students. Tal, et. al. (2005)

studied science teachers in urban settings that were attempting to implement PBL, and they

found that when teachers started feeling overwhelmed by the many expectations and the lack

of resources needed to live up to the expectations, they turned back to whole class teacher-

centered instructional techniques. This phenomenon is also occurring in math classes and,

just like in Tal, et. al.’s (2005) study, it is negatively affecting student learning, contributing

to the continuation of math deficits, frustration, and dissatisfaction.

For teachers who decide to attempt to learn how to teach math using PBL, there are

very strong PBL frameworks to consult, such as the Gold Standard PBL Teaching Practices

5

and Design Elements (PBLWorks, 2019). There are also many strong math frameworks to

consult, such as the NCTM Mathematics Teaching and Learning Practices (Martin, 1998;

NCTM, 2014; NCTM, 2018). Teachers can find blog articles, videos, and example projects

on the PBLWorks (2019) website, but only a few are geared towards math teachers.

Additionally, there is one book written by math educators Fancher and Norfar (2019) that is

rich with great examples and descriptions of how to implement PBL in a math classroom

(grades 6-10), but teachers would need to purchase and read the book to access those ideas.

There is a need for a reliable framework that can be consulted when designing and

implementing PBL units in a math class, especially at the high school level. Given that math

teachers already have difficulty implementing PBL, it is unreasonable to expect them to be

willing and able to consult all of these resources to ensure that they are developing and

implementing quality projects. The goal of this paper is to research and analyze the

similarities and differences of the existing frameworks for these effective teaching practices,

and to use that research to develop a new framework specific to project-based mathematics

classrooms that accounts for those best practices.

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CHAPTER 2: REVIEW OF LITERATURE

In this chapter, literature related to PBL and mathematics education will be reviewed

in order to provide a background for the framework. It will start by exploring the literature

related to why educators would want to use PBL in the mathematics classroom, analyzing

recommendations for mathematical learning and empirical studies of how project-based

classrooms created environments in which those recommendations could be fulfilled. Then,

the review will discuss the challenges that researchers have found when attempting to

implement PBL and similar pedagogical techniques. The following two sections will discuss

the effective teaching practices that have been defined in the literature for teaching

mathematics and teaching in a PBL classroom, respectively. The final section of the chapter

will review the math-specific PBL literature, which is mostly comprised of Fancher and

Norfar’s (2019) book.

Motivation for Using PBL in the Mathematics Classroom

NCTM “advocates for high-quality mathematics teaching and learning for each and

every student” (2017, p.1), so their research-based recommendations are widely used by

math educators and math education professionals in order to make curricula and course

decisions. In their most recent publication, Catalyzing Change in High School Mathematics

(NCTM, 2018), they recommend that students are offered mathematics courses that “do not

limit their ability to continue studying mathematics but expand their professional

opportunities, become equipped to understand and critique the world, and foster in them joy

and an appreciation for the beauty of mathematics” (NCTM, 2018, p.19). NCTM (2018)

stated in this report that they found evidence that students who do not develop a deep

understanding of mathematical content are more likely to drop mathematics courses as soon

7

as they are given the chance. Condliffe, et. al. (2016) notes that PBL seeks to create space for

deeper learning of the content, so it is a pedagogical approach that could remedy this issue. It

also is, by design, a methodology that connects students to professional opportunities, both

within discipline and across disciplines, and prepares students to understand and critique

their world, so PBL certainly would be useful for reaching the goals set by NCTM

(PBLWorks, 2019; Grossman, et. al., 2019). The final goal, fostering joy and appreciation of

mathematics, seems to be the loftiest of the goals, generating doubt among math teachers

who have worked with a wide array of student demographics (S. Graham, personal

communication, January 18, 2021). This paper will show, however, that learning how to

teach math using PBL empowers teachers to create learning experiences where students

begin to understand just how beautiful and powerful mathematics can be.

Grossman, et. al. (2019) surveyed experts in PBL and studied teachers that had been

identified as effective project-based practitioners to learn what teaching practices were

necessary to implement a good PBL unit. One of the practices that they identified was that

teachers "engaged students in disciplinary practices” (Grossman, et. al., 2019, p.45). This

means that math teachers should be ensuring that students act as mathematicians would,

practices which are outlined for teachers in the standards for mathematical practices

associated with CCSS (NCTM, 2014). As Knuth (2000) points out, in his study on how

students understand mathematical connections between graphs and equations, experts must

have knowledge that “extends beyond simple procedural competence,” (p.506), so it is not

enough for students to master a procedure involved in solving a problem. Instead, they must

be able to move flexibly between representations and to choose appropriate representations

for solving a variety of problems (Knuth, 2000). PBL is a classroom strategy that supports

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this development of flexible understanding, since PBL is founded on authenticity in

disciplinary practices as well as real world connections (Grossman, et. al., 2019). In PBL,

students are presented with the opportunity to reason through challenging problems

(PBLWorks, 2019), author ideas, and justify their reasoning. In her work on collaborative

learning in the mathematics classroom, Langer-Osuna (2017) found that students who author

their own mathematical ideas and are able to justify their reasoning tend to form more a

robust understanding of the mathematical concepts associated with procedures that they are

expected to learn. Similarly, Herbel-Eisenmann, et. al. (2013) recommended six teacher

discourse moves, all of which foster student discourse around mathematical ideas, because

they found that when students talk about math, they are more likely to learn math. It stands to

reason that math teachers would benefit from learning how to implement PBL in their

classrooms, since PBL is capable of fostering a classroom environment in which students

engage in the learning opportunities that NCTM recommends and studies support.

Fancher and Norfar (2019) and Condliffe, et. al. (2016) both point out that as society

changes, mostly due to technological advances, the role of school, and more specifically

mathematics, changes as well. Calculators and computers can now do all of the procedural

work that used to fall on the mathematician, so now it is more important for students to be

able to utilize mathematics to make sense of a problem and create models that could help

solve the problem (Grossman, et. al., 2019). School used to prepare students to work in

factories, sit in lecture halls taking notes, and live in a society where following rules was one

of the most important aspects of being a citizen. Now, being college and career ready means

that students have developed “success skills, such as critical thinking, self-regulation, and

collaboration” (Condliffe, et. al., 2016, p.6). These skills are also described in NCTM’s

9

(2018) effective mathematical practices and are key elements of the PBL model (PBLWorks,

2019). Thus, advocates for mathematics education should turn to PBL as a primary resource

for developing curricula.

Barriers to PBL in the Mathematics Classroom

There are a number of barriers that teachers list when describing why they are

hesitant to use project-based learning approaches in their classrooms. Haag and Megowan

(2015) studied a similar teaching approach, inquiry-based learning (IBL), in science

classrooms to learn more about teacher readiness in implementing IBL. One of the most

prominent barriers they found was time. It is reasonable to conclude that math teachers

attempting to use PBL would note similar issues, in fact, Aslan and Reigeluth (2015) found

that math teachers struggled with PBL because the pace of the state standards was too fast.

DiBiase and McDonald (2015) elaborated on this barrier in their study on science teacher

attitudes towards IBL, finding that the time constraint had many facets including days in the

semester, time allotted to each class period, and time to teach the number of standards

allotted per course. Again, these issues are not unique to science courses, and certainly affect

the decisions that math teachers make each day about how they will introduce students to a

concept. In addition to concerns about having enough time with the students for PBL,

teachers also have to come to terms with the amount of time they will need to spend

“providing feedback, guiding reflective activities, and helping students consider how they

can improve their work” (Grossman, et. al., 2019, p.47). This can be especially problematic

in large class sizes, which DiBiase and McDonald (2015) and Tal, et. al. (2005) both note

when investigating teacher concerns about time. Teachers also must find time to attend

professional training to develop their PBL skills and their content knowledge (Haag &

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Megowan, 2015), since teaching using PBL “requires expert application of knowledge and

constant adaptation to diverse contexts and students” (Grossman, et. al., 2019, p.48).

Managing all of these time constraints requires teachers to creatively redesign the way they

approach their classes, which is often too much for teachers in the midst of an already hectic

job. Thus, advocates seeking to support math teachers in implementing PBL must help them

understand how to utilize PBL as a solution for learning concerns, not an added burden.

Another prominent barrier to effectively using PBL to teach math is teacher content

knowledge. As mentioned previously, Grossman, et. al. (2019) found that effective PBL

teachers must be capable of applying expert content knowledge when developing their lesson

plans in order to create opportunities for students to engage in disciplinary practices.

Additionally, teachers must be able to collaborate with students on ideas that surface during

the lesson, regardless of whether those ideas had been thought about during the planning

phase (Grossman, et. al., 2019). Teachers that do not have a deep grasp of their content and

the disciplinary practices associated with it will have difficulty designing these opportunities

as well as recognizing and encouraging them when their students discover something that

they do not completely understand. Authenticity is also a key element of PBL (PBLWorks,

2019; Grossman, et. al., 2019), and expert mathematical knowledge is essential to being able

to find authentic connections between mathematical disciplines. Examples of these

connections include: how statistics or geometry can be used to develop algebraic ideas, how

mathematics is used in other disciplines (NCTM, 2018), how mathematics can be used to

solve a global problem (Himes, et. al., 2020), or how personal experiences in students’ lives

can influence their solution to a mathematical problem (Fancher & Norfar, 2019). It can seem

daunting to look at the long list of mathematical standards and attempt to find authentic

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connections that can elicit sustained inquiry in a project, but Fancher and Norfar (2019)

maintain that, once a teacher starts this process, it becomes challenging to look at a standard

or a situation and not see a project opportunity.

Finally, according to DiBiase and McDonald (2015), teacher beliefs on how to best

prepare their students for standardized assessments prevents many teachers from using PBL

in the classroom. They found that there is a lot of pressure placed on teachers to prepare their

students to perform well on the standardized state assessments. This leads teachers to believe

that it is more important to develop procedural fluency and test-taking skills than it is to

develop problem solving skills or conceptual understanding. Teachers are given a long list of

standards to teach their students, most of which will be assessed by at least one question on

the test, and they believe that time spent on things like PBL will take away from the time

students need to become fluent in the procedure that will be tested (DiBiase & McDonald,

2015). The standards have been rewritten in an attempt to encourage teachers to spend more

time on problem solving, interpretation, and model building (Condliffe, et. al., 2016; Confrey

& Krupa, 2011), but the state assessments do not seem to support these adjustments in the

standards (DiBiase & McDonald, 2015). Therefore, many teachers are still resistant to

exploring innovative practices such as PBL that they believe will take away from their

students’ ability to score well on the assessment.

There will always be reasons for teachers to avoid learning how to teach using a

different tool. However, in the effort to provide quality mathematics education to each

student, it is important to learn how PBL can be utilized in the mathematics classroom and to

strive to provide support to teachers as they implement PBL in their own classrooms.

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Effective Mathematics Teaching Practices

Over the past few decades, NCTM (Martin, 1998; NCTM, 2014; NCTM, 2018) has

been advocating for students to get the opportunity to solve problems with a variety of

appropriate strategies, communicate their mathematical reasoning, and create representations

to model and interpret phenomena in mathematics and other disciplines. They have released

short lists of teaching and learning practices for teachers to utilize, such as the “5 Process

Standards” (Martin, 1998, Figure 2.1) and the “8 Mathematical Teaching and Learning

Practices” (NCTM, 2014, Figure 2.1). This section will briefly explain these different

frameworks and then chapter three will go into greater detail about each component and how

they connect to the other frameworks.

Figure 2.1

Teaching and Learning Practices

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Each of these lists encourage teachers to create an environment where their students

are actively making sense of problems and utilizing mathematical models to find solutions

(NCTM, 2014). NCTM (Martin, 1998) asks teachers to give students the opportunity to

communicate their reasoning with their peers, author ideas that could be used to solve the

problem, and reflect on their models and solutions in order to make adjustments.

There have been a variety of researchers seeking to help teachers understand how to

create this environment. In his investigation of how to support productive struggle in learning

math, Kapur (2013) found that teachers must begin their instruction by giving students a

challenging task. Too often, students are presented with multiple tasks that follow a

prescribed solution that they have already been taught, and they fail to develop the ability to

flexibly apply and adapt their problem-solving strategies (Knuth, 2000; Martin, 1998). For

example, as Knuth (2000) describes, students may learn about lines by looking at equations

and graphs in the context of slope-intercept form. Then, they are presented with multiple

problems, some word problems, some graphing problems, but all with the exact solution path

that they have recently been taught, using slope-intercept form. There is little flexibility in

how they might solve the problem, there is no room for creativity in solving the problem, and

the main struggles that students might face are related to misconceptions rather than the

nature of a messy problem. Heibert and Carpenter (1992), in their study of how

mathematical language supports mathematical understanding, were concerned about this

because they saw that students who were taught to simply repeat procedures that they had

learned often failed to draw the mathematical connections across representations, despite the

obviousness of those connections to their teachers. In order to address this concern, NCTM

(2014) recommends that students experience productive struggle while working on a task. In

14

response to this recommendation, Warshauer (2015) studied strategies that math teachers use

to support productive struggle. He contends that students should be allowed to experience

struggle, test their thinking, and oscillate between doubting themselves and believing in

themselves as they work through a problem and apply mathematical ideas (Warshaeur,

2015). This process does take time and requires the teacher to create an environment where

mistakes are celebrated as learning opportunities (Fancher and Norfar, 2019), but it is how

mathematicians (and other scientists) engage in a problem, so it is important for students to

experience it (Grossman, et. al., 2019). It also allows teachers to give their students agency

over their learning, and share the authority of mathematical understanding with their teacher

and peers (Langer-Osuna, 2017). When students get the opportunity to engage in this

productive struggle through a challenging task that could be solved in multiple ways, then

they are more likely to be able to make meaningful mathematical connections (NCTM,

2014).

PBL not only supports all of these suggested teaching and learning practices, but it

also provides more details for how teachers can create the environment in which students

have these opportunities. Fancher and Norfar (2019) outlined four teaching practices that

they believe are important for using PBL in the mathematics classroom: “plan lessons that

are standards-based, encourage wonder and curiosity, provide a safe environment in which

failure occurs, and give students opportunities for revision and reflection” (p.8). These

practices are exactly the same practices that NCTM has been advocating for all of these

years, which demonstrates that PBL absolutely can be used in the math classroom and could

even be one of the best pedagogical methods to support the effective mathematics teaching

practices.

15

Effective Project-Based Teaching Practices

There are two frameworks that will be used to understand what effective project-

based teaching practices look like: PBLWorks’ (2019) Gold Standard PBL (Figure 2.2) and

Grossman, et. al.’s (2019) Core Practices of Project-Based Teaching (Figure 2.3). PBLWorks

(2019) separate their Gold Standard PBL into teaching practices and design elements, where

the teaching practices describe the culture of the classroom and the design elements describe

the project itself.

Figure 2.2

PBLWorks Gold Standard Design Elements (left) and Teaching Practices (right)

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Figure 2.3

The Core Practices of Project-Based Teaching

The environment necessary for effective PBL is very similar to the environment

described earlier for effective mathematical learning. According to the PBLWorks (2019)

Gold Standard PBL framework, teachers design and plan a project centered around a

challenging problem that is aligned to the relevant content standards and promotes

perseverant problem solving. Each aspect is important; if the problem is not challenging, then

students will not be able to sustain inquiry, and if the project is not aligned to relevant

content standards, then the time and effort will not seem worthwhile. Teachers need to build

a culture that supports student inquiry and agency, which includes teaching them how to

regularly reflect and revise (PBLWorks, 2019; Grossman, et.al., 2019). Students should be

actively exploring solutions, learning content that helps them make sense of the problem, and

17

go through an iterative process of reflection and revision until they have created the best

version of their solution possible (within the time constraint, of course).

Teachers share the authority of learning with students by engaging in student

reasoning, assessing student learning, and coaching students as they work to reach their

project goals (PBLWorks, 2019). Since teachers cannot plug their brains into the brains of

their students and download the relevant information, it is essential that students are active

participants in their own learning. It is equally important that teachers remain aware of

student learning and student goals, so they can continue to support students, hence the need

for continuous formative assessment. This collaboration between the teacher and the students

gives the students ownership of what they learn, what they need to learn, and how they plan

on learning it, with the teacher acting as a facilitator.

Finally, PBLWorks (2019) states that project-based teachers provide students with

authentic learning experiences, where they are able to engage in disciplinary practices, create

products for authentic audiences, and make sense of problems set in a real-world context.

Authenticity, according to Grossman, et. al. (2016), is a key ingredient for student motivation

since it allows students to make connections between what they are doing in math class either

to things that they care about outside of school or to why anyone in the world might care.

PBLWorks (2019) uses the phrase public product to describe the concept of an authentic

audience, because they want teachers to think about how students can present their findings

publicly. This is important for motivating students to produce quality work, as Revelle, et. al.

(2019) found after looking at the writing of second graders when asked to write for a public

librarian rather than their teacher. According to Spires, et. al. (2019), in their study of

interdisciplinary IBL, in addition to having a final public product, the authentic audience can

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be accomplished throughout the project by incorporating external experts into the feedback

process, further motivating reflection and revision. Fancher and Norfar (2019) suggest that

teachers start small, creating authentic situations for external adults to role play for students,

such as acting as clients, and then when they are comfortable with that, teachers can begin to

branch out to community members. Fortunately, communities are full of people who would

love to help students learn, such as university professors, public policy makers, school and

district leadership, and local small businessmen. Additionally, as Spires, et. al. (2017) found,

technology has made it possible for students to connect with communities worldwide, so

consulting with experts has become even easier because experts can work with students via

the internet.

Much of what has been described could simply be called great teaching, as there are

many great teachers that are creating rich, student-centered learning environments without

the use of projects (Grossman, et. al., 2019). However, as Grossman, et. al. (2019) goes on to

say, project-based teachers leverage the project context to drive sustained rigorous inquiry,

which allows students to ground their learning in a compelling context. This helps them to

make connections and retain learned concepts and procedures, as well as increases

motivation. Condliffe, et. al. (2016) makes sure to point out that motivation is a happy by-

product of PBL, and should not be the main draw or the deciding factor when choosing a

project context. The project is the central driver of the learning goals, Condliffe, et. al. (2016)

states, and should span a significant period of time in which multiple standards are addressed

in connection to the project. As Spires, et. al. (2017) points out, the project should not be

extra, but rather integrated strategically into the curriculum, serving as the conduit for

learning the desired standards. PBL does motivate students to produce quality work by

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allowing students voice and choice in their project designs and finding authentic audiences

that will experience the final product or provide helpful critiques along the way (PBLWorks,

2019). More importantly, it teaches students how to continually seek improvement while

time allows, accept feedback from teachers, peers, and external experts, and apply that

feedback productively to create a better iteration of their product or solution (PBLWorks,

2019, Grossman, et. al., 2019). PBL is great teaching, but it is also so much more, and thus

math teachers who take the time to learn how to implement PBL in their classes will

experience learning rewards beyond what they knew was possible.

Using PBL in a Mathematics Classroom

According to Condliffe, et. al. (2016), “it has been noted that math teachers have

found it [particularly] difficult to integrate PBL into their instruction” (p.iii). This is

somewhat surprising given how much similarity there is between the suggested teaching

practices for math and PBL. On the other hand, maybe this is not so surprising after all. Math

teachers have been struggling to implement the recommended teaching practices for

mathematical learning, Heyd-Metzuyanim, et. al. (2018) assert in their study of math teachers

learning from professional development around the five practices for orchestrating

mathematical discussions. Jung and Newton (2018) found, in their study of pre-service

teachers’ understanding of mathematical modeling, that this is most likely due to differing

interpretations of the practices. They assert that math teachers are likely to interpret the

practices by referencing their experiences learning math, so if they have not experienced

mathematical modeling in the way that researchers intended, or they have not seen examples

of it, then they will instead think about how they learned math when they were in school or

how they have taught it in the past (Jung & Newton, 2018). Heyd-Metzuyanim, et. al.’s

20

(2018) work confirms this claim, stating that teacher beliefs tend to change after they have

implemented a new idea and witnessed the results firsthand. Fortunately, Fancher and Norfar

(2019) provided some great examples of how to design PBL units for the math classroom,

and this paper provides more examples to help explain each aspect of the framework. Keep in

mind, however, that the research suggests that teachers will need professional development

and support that model these examples or others in order to truly believe that PBL is a good

model for teaching and that they will be capable of utilizing it (Heyd-Metzuyanim, et. al.,

2018).

Their first suggestion is to simply invert the traditional methods of mathematical

teaching. Introduce the concept with a word problem, stripped of specific numbers and rich

with potential for curiosity (Fancher & Norfar, 2019). For example, consider this textbook

problem: find the surface area and volume of a Campbell’s soup can that has a diameter of 3

in. and a height of 10 in (Fancher & Norfar, 2019). Most textbook problems include the

numbers like this for students, which allows students to identify the relevant formula that

they have learned how to use, plug in the numbers, and, voila, the problem is solved. Instead,

as Fancher and Norfar (2019) suggest, the goals of this problem can be met by having the

students act as marketing consultants to Campbell Soup. There are multiple things that

Campbell Soup wants to change about their cans: the label is tearing on the edges, so they

need students to design a label that can cover the can but leave enough room to tear, and the

soup is sloshing over the edge when the machine puts it into the can, so the students need to

figure out exactly what volume of soup to tell the machine to place into the cans. With these

problems, students might explore different size cans that Campbell Soup offers, and then

figure out the solutions to these issues for those different sizes. Adjusting textbook word

21

problems is a great strategy for beginners of PBL because math textbooks and resources are

full of word problems with contexts that are relevant to students (Fancher & Norfar, 2019).

Fancher & Norfar (2019) do want teachers to keep in mind that not all word problems are

rich enough to be adjusted this way, so there is some strategy to selecting a word problem to

turn into a project, but many problems are rich enough so this is a great place to start. The

power of stripping the numbers from the problem is that it allows students to develop a

conceptual and contextual understanding, and experience the need to know for the associated

mathematical procedure, which is something Condliffe, et. al. (2016) found to be integral to

successful PBL lessons.

As students explore the problem, Fancher and Norfar (2019) advise that teachers

strategically anticipate what students will need to know and design tasks that help them build

those skills. Similarly, Smith and Stein (2011), in their five practices of scaffolding student

learning in an inquiry-based math classroom, encourage teachers to anticipate likely student

solutions to the challenging task at hand. Each learning task experienced during the PBL

process should create curiosity around important learning goals for the project, so that

students can continue to learn the standards that are aligned (Fancher & Norfar, 2019). In the

Campbell Soup project, for example, Fancher and Norfar (2019) had a lesson planned for

students to explore various sizes and shapes of cylinders and learn how to calculate the

volume and surface area of those cylinders. They anticipated that this learning task would

help students learn about what they needed to know from the client, such as height and radius

of the cans, in order to solve the problem presented to them. Throughout the project, students

should be developing the skill of identifying areas where they need to learn more, and

seeking the resources that can teach them those skills (Fancher & Norfar, 2019; Grossman,

22

et. al., 2019). Fancher and Norfar (2019) describe how they have their students write down

things they still need to know, and how they will learn them, with ‘from the teacher’ only

allowed once per week of the project. As mentioned previously, it is essential that the project

drives the learning that occurs, and simultaneously the learning that needs to occur should

drive the project (Condliffe, et. al., 2016). Thus, the learning tasks need to be intentionally

placed so as to focus student learning after their curiosity has run wild, without stifling

student curiosity by teaching concepts prematurely. Condliffe, et. al. (2016) further explains

that the work on the project should not be peripheral to the mathematical learning goals, but

rather a reason to be learning that mathematics.

Some math teachers find it challenging to figure out what the public product or

authentic audience could be for a project. Fancher and Norfar (2019) recognize that, so they

suggest that teachers utilize the concept of a client that students must design for, which is

what they do in most of their projects. The purpose of the project is authentic by reflecting

scenarios that occur regularly in career settings, and then they bring in outside adults to play

the role of the client (Fancher & Norfar, 2019). Whenever possible, it is nice to find adults

that can play this role who actually work in a related field, such as bringing in family

members, friends, or neighbors. In the example of the Campbell Soup cans, it would be

useful for students to meet with marketing professionals or manufacturers, to understand the

importance of the labels and sloshing soup to the company. These professionals would also

be able to provide insight into potential issues with proposed student solutions, which is why

attempting to connect with related experts is so powerful. However, teachers can also do

some research around the ideas that they are presenting to provide to the ‘client’ actors so

students can still experience this type of feedback. Other PBL advocates, such as Himes, et.

23

al. (2020) in their study of interdisciplinary IBL, suggest that teachers can find local and

global experts in a related field by utilizing universities, parent communities, and local

policymakers. Again, this, as Spires, et. al. (2017) pointed out, has become especially easy to

do with the various communication technology, such as email and videoconferencing, that

exists today. Working with experts to solve part of a problem to which they have devoted

their careers creates an even more authentic level to the work students are doing (Himes, et.

al., 2020). Additionally, Himes, et. al. (2020) has found that connecting student work with an

appropriate social contribution, such as raising money for a nonprofit related to their

exploration, also creates authenticity for them. Grossman, et. al. (2019) also speaks to this

idea of making a contribution, finding that many PBL teachers figured out ways for student

products to change things at their schools or in their local communities, such as building a

community garden. These ideas, although often interdisciplinary, all have the potential to

have connections to mathematics standards with a little creativity. The more projects that

teachers complete, the more they will see connections to their standards in the world around

them, and they will become more comfortable leveraging the relationships that they make in

their communities to support student learning (Fancher & Norfar, 2019).

In addition to finding an authentic audience, teachers can create other authentic

connections to the experience, even if the public product is more of a role play or

presentation within the school community. One thing that teachers can do is learn about the

personal interests of their students and incorporate those interests into project choices

(Grossman, et. al., 2019). For example, in Mr. Smaldone’s class (personal communication,

March 10, 2021) students were given the opportunity to create art by using technology to

graph polynomial functions. In this example, students were allowed to choose what image

24

they might like to create, which gave artists the space to be more creative while other

students found images that are traceable but still representative of their interests. Another

thing that teachers can do is learn more about the associated disciplinary practices, such as

how certain mathematical standards are utilized by actual mathematicians, scientists,

economists, designers, or engineers, to name a few (Grossman, et. al., 2019). The Campbell

Soup project is an example of this, because it draws on the mathematics involved in the daily

decision-making that marketing professionals and manufacturing engineers might do on a

daily basis. Teachers could also turn to history, as Bressoud (2010) suggests in his reflection

on the history of teaching trigonometry, as well as the various mathematical disciplines

(algebra, statistics, geometry, etc.) to learn how to create an authentic learning experience. In

other words, why was the procedure or convention invented, what problem was it attempting

to solve? How can that problem be recreated for students? Additionally, how could an

algebraic concept be derived from something related in statistics or geometry? For example,

statistical regression models can be used to fit equations to a scatterplot of the data from a

data set pulled from the internet, such data about clean drinking water available to people in

different countries around the world. This would provide context to the equation in an

authentic way, since most models in the real world are not based on perfect situations. After

fitting a model, it then makes sense to learn about the different features of that model, such as

the domain and range or intervals of increasing and decreasing, because those features can be

used to communicate with an audience what the graph represents in the context of interest.

Similarly, the history of mathematics is ripe with authentic experiences for commonly used

procedures and conventions taught in math classes today, so that is another way that students

can develop the need to know that drives their learning (Bressoud, 2010). An example of this,

25

according to Bressoud (2010), is to recreate the experience of the astronomers who first

developed trigonometry as a connection between arcs and lengths in a circle in order to

understand the movement of the stars and planets in relation to the earth. Throughout history,

people made discoveries like this that required the development of new mathematical

structures, so projects made to recreate these experiences could help students understand and

appreciate the beauty of mathematics.

Math teachers can implement PBL in their classrooms and address the many state

standards that students are expected to learn in one course. In fact, PBL should not be done

unless it is addressing the mathematical learning goals of the course. There are a variety of

ways to develop PBL units for a math classroom, and once math teachers get started down

the path, they will have a hard time looking at a standard or a situation and not seeing a

project opportunity. Although it can be challenging to learn in the beginning, the rewards that

come from watching students develop deep understanding and appreciation of mathematics

make it worth the effort. In the next chapter, this paper has merged all of the discussed PBL

and mathematics frameworks into one framework that can further support math teachers on

their journey to implementing PBL in their classrooms.

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CHAPTER 3: DISCUSSION OF THEORETICAL FRAMEWORK

It is now clear that PBL should be used in math classrooms, and that math teachers

are not yet comfortable teaching using PBL with the resources that exist. The first section of

this chapter will draw explicit connections between the different frameworks that were

discussed in the two sections on effective teaching practices from the previous chapter. The

following section will describe the new framework and show how it captures the heart of

each of the previously discussed frameworks in one resource for math teachers. This section

will then explain each aspect of the new framework in detail and provide an example of how

it would look in a project.

Connecting the Effective Teaching Practices

The various frameworks discussed in the effective teaching practices sections of the

literature review have strong connections to each other. For ease of description, each

paragraph will begin with Grossman, et. al.’s (2019) Core Practices of Project-Based

Teaching framework (Figure 3.1) and draw connections accordingly.

Core Practice 1: Disciplinary

Grossman, et. al.’s (2019) framework begins with the core practice of being

disciplinary. They outline three subcategories of this practice: “elicit higher order thinking,

orient students to subject-area content, and engage students in disciplinary practices”

(Grossman, et. al., 2019, p.45). The following paragraphs will draw explicit connections

between the three parts of this core practice and the different aspects of the other frameworks

(Figures 3.2 and 3.3; Table 3.1) described in previous sections.

27

Figure 3.1

Grossman, et. al.’s (2019) Core Practices of Project-Based Teaching Framework

Figure 3.2

Gold Standard PBL: Design Elements (left) and Teaching Practices (right)

28

Figure 3.3

NCTM Effective Mathematics Teaching Practices

29

Table 3.1

Smith and Stein’s (2011) Five Practices for Orchestrating Productive Mathematical

Discussions

Five Practices for Orchestrating Productive Mathematical Discussions

1. Anticipating likely student responses to challenging mathematical tasks and

questions to ask to students who produce them.

2. Monitoring students’ actual responses to the tasks (while students work on the

tasks in pairs or small groups).

3. Selecting particular students to present their mathematical work during the whole-

class discussion.

4. Sequencing the student responses that will be displayed in a specific order.

5. Connecting different students’ responses and connecting the responses to key

mathematical ideas.

Grossman, et. al.’s (2019) practice of eliciting higher order thinking is also a

mathematical teaching practice outlined by NCTM (2014, Figure 3.3), using the wording

“elicit evidence of student thinking” (p.10). It is also strongly aligned with the PBLWorks

(2019) teaching practice (Figure 3.2) of scaffolding student learning. Teachers must prepare

questions and implement tasks (NCTM, 2014, Figure 3.3) that enable students to engage in

higher order thinking, as well as practice the art of “posing purposeful questions” (NCTM,

2014, p.10, Figure 3.3) when they encounter unexpected student thinking. In order to become

more adept at this practice, Smith and Stein (2011) outlined five practices for scaffolding

student learning in an inquiry classroom. The first two are anticipating likely student

30

responses and monitoring students’ actual responses (Table 3.1), which are important in

student-centered learning because teachers must be capable of engaging with student

reasoning. Teachers who are regularly thinking about potential approaches students will take

when introduced to a challenging problem, and then actively learning about their students’

actual approaches as they work on solving a problem, are more likely to be able to pose

purposeful questions (Smith & Stein, 2011). Expecting students to “use and connect

mathematical representations” (NCTM, 2014, p.10, Figure 3.3) and providing opportunities

for students to engage with multiple connected mathematical representations also allows for

teachers to elicit higher order thinking from their students during class. Thus, the practice of

eliciting high level thinking from students is intertwined throughout the effective teaching

practice frameworks for PBL and mathematics.

The second subcategory of Grossman, et. al.’s (2019) disciplinary practice is to orient

students to subject-area content. In the PBLWorks (2019) teaching practices seen in Figure

3.2, this is called aligning to standards, and Fancher and Norfar (2019) use the words

“planning lessons that are standards-based” (p.7). In NCTM’s (2014) teaching practices in

Figure 3.3, they name this practice “establishing mathematical goals to focus learning”

(p.10). It is evident from each framework that a project must be rooted in the learning goals

of the class, or it is not considered PBL. This is important because PBL is considered to be a

pedagogical tool, where the project is central to the learning rather than simply an assessment

of the learning that has taken place, often referred to as ‘dessert projects’ (Condliffe, et. al.,

2016). Fancher and Norfar (2019) suggest that the best projects usually begin with a look at

the standards and then layer in the authenticity where they see inspiration in the world that

connects to the learning goals stated in the standards. For example, in the polynomial art

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project described earlier, students could potentially want to use circle equations, but if the

content standards for the course do not address circle equations, then the teacher should be

aware of how to support students in creating something that does not rely on circle equations,

but rather demonstrates knowledge of the relevant polynomial standards in the course. This

might involve setting requirements for the students of the types of functions that they are

allowed to use, and providing feedback to student designs along the way when they are not

aligned with the learning goals. There are many interesting ways to use mathematics in the

world, but remember that Condliffe, et. al. (2016) cautions teachers that motivation is an

awesome side effect of PBL, not the reason to use it. PBL is only effective if it is leveraged

to teach students what they need to learn to be successful in the course.

The final subcategory of Grossman, et. al.’s (2019) disciplinary practice, engage

students in disciplinary practices, is colored purple in the diagram (Figure 3.1) because it also

provides authenticity to the project. Thus, it is connected to the PBLWorks (2019) design

element of authenticity. In mathematics, disciplinary practices include designing and testing

mathematical models in order to solve a problem, which means this category is connected to

the NCTM (2014) teaching practices of using and connecting mathematical representations

and “implementing tasks that promote reasoning and problem solving” (p.10). Additionally,

mathematicians, much like most practitioners, must be able to communicate their findings to

the mathematics community. This means the NCTM (2014) practices of eliciting evidence of

student thinking and facilitating mathematical discourse allows teachers to support student

development of mathematical communication. Since mathematicians contribute to the

mathematical body of knowledge by continually wondering about what else could be learned,

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Fancher and Norfar’s (2019) practice of “encouraging wonder and curiosity” (p.7) is also

connected to this aspect of the disciplinary practice.

Core Practice 2: Authenticity

The next core practice in Grossman, et. al.’s (2019) framework (Figure 3.1) is

authenticity. There are two subcategories in this practice, in addition to the one previously

discussed: “support students to build personal connections to the work and support students

to make a contribution to the world” (p.46). This is a popularly discussed practice associated

with PBL, and has proven to be particularly challenging for math teachers, especially those

teaching higher level mathematics (Condliffe, et. al., 2016). PBLWorks (2019) has the exact

same core practice included in their design elements framework (Figure 3.2) because they

also believe it is one of the most important pieces of PBL.

In the math class, teachers can support students to build personal connections to the

work (Grossman, et. al., 2019) by eliciting evidence of student thinking, using and

connecting mathematical representations (NCTM, 2014) and encouraging wonder and

curiosity (Fancher & Norfar, 2019). When engaging with students about their thinking,

teachers should not only wonder about a student’s understanding of the mathematics that

they are exploring, but also about how they are making sense of the problem in context of

their previous experiences. Mathematical representations are most powerful when connected

to a context, which gives math teachers the opportunity to design projects that capitalize on

student experiences to build their models. For example, in Fancher and Norfar’s (2019)

project simulation, students were introduced to a client that runs a movie theater and needs a

new popcorn container that appears to hold lots of popcorn, but actually holds much less. In a

situation like this, students are engaging with mathematical representations that will help

33

optimize this client’s popcorn container, while simultaneously drawing upon their personal

experiences with buying and eating popcorn (Fancher & Norfar, 2019). Students will also

naturally make connections to their personal lives if teachers encourage them to follow their

curiosities as they explore the problem (Fancher & Norfar, 2019). Teachers who regularly

converse with students about their interests for social emotional purposes may also find that

they can draw upon these conversations when designing projects, and know that it will

certainly be personally connected to at least one student in the class. This practice also helps

teachers avoid situations in which they think a context will be relevant to students, such as a

quadratics roller coaster project, only to find out that none of their students care about roller

coasters and would have rather designed the animations necessary for a videogame character

to throw objects across the screen (N. Smaldone, personal communication, February 22,

2021). Since math is often seen as disconnected from personal experiences for students, it is

important for teachers to integrate relatable contexts into mathematical learning experiences.

Supporting students to make a contribution to the world (Grossman, et. al., 2019) can

seem a little daunting to teachers. This is especially true for math teachers who are often

bombarded by students and parents doubting its usefulness beyond the classroom, as

Kilpatrick and Izsak (2008) described in their historical account of algebra in school

curriculum. This is where the public aspect of the PBLWorks (2019) design element public

product (Figure 3.2) comes into play. As described before, Fancher and Norfar (2019)

usually utilize fictional clients and bring in outside adults, who preferably experience those

roles in their careers, to play those fictional clients. For teachers looking for an even more

powerful way to connect students to an authentic audience, they could find community

members with authentic needs related to the project and have students design on their behalf.

34

For example, students could meet with a local entrepreneur who is in need of a logo and

compete to design the best logo using aesthetic principles related to geometry, such as

symmetry. It can also be done by finding causes or organizations to support that are doing the

work that students are studying, like in the project done by Himes, et. al. (2020), where

students donated water to a local food bank after studying how access to clean water affects

people locally and globally. Himes, et. al. (2020) also connected students to an expert in

global sanitation through the local university, leveraging the internet and the passion of a

university professor to generate interest and knowledge amongst the students. Spires, et. al.

(2017) connected students from the United States with students from China to do a project

comparing their cultures, and one could imagine that students could learn a lot from each

other mathematically as well in a partnership like that. Additionally, NCTM’s (2014) practice

of mathematical modeling can, and should, be leveraged to support students’ ability to

communicate with an audience. An example of this might be in the case of students

proposing to build a community garden at their school. In order to get the proposal approved,

they would need to present mathematical models that help their principal know how much

the garden will cost to build, how much it will cost to run, and how they plan on funding

those costs. Maybe they will be seeking funding from an outside organization, so they will

need to take their presentation to that organization as well. These models would need to be

visually appealing and easy to understand to people who have not been studying the

mathematical ideas. They might also need to diagram how much space the garden will take

up, as well as the design of the garden plots, relying on geometrical knowledge for spatial

awareness. Students feel empowered when their work at school serves an authentic purpose

in their communities, so these opportunities have a lasting effect on student learning.

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Although not included in Grossman’s (2019) framework, the challenging problem or

question in the PBLWorks (2019) framework (Figure 3.2) is a good way to create

authenticity for students from the beginning. With a compelling question or problem,

students become naturally curious, making it easy for teachers to encourage wonder and

curiosity (Fancher & Norfar, 2019). This also aligns with NCTM’s (2014) suggestion to

implement tasks that promote reasoning and problem solving. Keep in mind, however, that

the majority of the mathematical learning tasks described by NCTM (2014), Smith and Stein

(2011), and Boaler (2016) are more useful for daily lesson plans found throughout a PBL

unit, rather than serving as the PBL unit itself. The challenging problem or question,

alternatively, is intended to overarch each lesson plan, with the tasks connecting to the goals

of the question, and should extend over a much longer period of time (Condliffe, et. al.,

2016). The daily tasks can then be used to support the sustained inquiry work throughout the

project, serving to address student need-to-knows, assess student learning, and ground

student solutions in relevant learning goals. For example, in the popcorn container project,

Fancher and Norfar (2019) had their students use clay one day to create different container

shapes and investigate the volume of those shapes and how to determine if they appeared

larger than their actual volume. This would be considered a challenging mathematical task

because it has many different solutions for students to come up with, but it is only one lesson

in the overarching project of designing a container for the client. Other necessary math

lessons in this project might include surface area of the containers for material cost purposes

and how to determine pricing options of the popcorn. It should be impossible to answer the

challenging question in just a few days, and should lend itself to the ability to continually

improve with feedback throughout the course of the project. If students are capable of finding

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a perfect solution on the first day, or uninterested in improving their initial solution, then the

question or problem is not challenging or compelling enough.

Core Practice 3: Iterative

The third core teaching practice in Grossman, et. al.’s (2019) framework is that the

process is iterative, which they explain using three subcategories of what teachers do: “track

students’ progress and provide feedback, support students to give and receive feedback, and

support students to reflect and revise” (p.47). The iterative nature of the process is very

important to PBLWorks (2019). It motivated the circular representation of the Gold Standard

PBL frameworks (Figure 3.2), and the subcategories of Grossman et. al.’s (2019) iterative

category are very similar to two of the Gold Standard design elements and one of the Gold

Standard teaching practices.

Tracking student progress and providing feedback (Grossman, et. al., 2019) is an

important teaching practice in all pedagogical methodologies, but does not always have an

iterative nature. In the PBLWorks (2019) teaching practices framework, they call this process

assessing student learning, but in their design elements framework, they use the phrase

critique and revision, taking care to include time for students to revise based on the feedback.

In Smith and Stein’s (2011) practices for supporting mathematical discourse, they refer to

tracking student progress (Grossman, et. al., 2019) as “monitoring students’ actual responses

to the task” (p.10). In other words, formative assessment is crucial to effectively teaching

PBL in the math classroom. Throughout each class, teachers need to be actively collecting

evidence of student understanding so that they can assess how students are progressing in

terms of the mathematical goals as well as in finding a solution to the challenging problem,

and then determining what tasks need to be implemented in subsequent lessons. Smith and

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Stein (2011) also provide strategies for math teachers to be able to provide feedback during

class time: “select particular students to present their mathematical work during whole-class

discussion, sequence the student responses that will be displayed in a specific order, and

connect different students’ responses [to each other] and to key mathematical ideas” (Smith

& Stein, 2011, p.10). Basically, in the process of presenting their work and making explicit

connections, students will receive feedback from their peers and their teacher and be able to

revise any misconceptions that may have occurred in the first iteration of the solving process.

Feedback loops are very important for student learning, Ferriter and Cancellieri (2017) state

in their book on creating a culture of feedback in the classroom. Students need feedback to

know if they are on the right track or have made a mistake that is affecting their process

(Ferriter & Cancellieri, 2017). However, this feedback does not have to come directly from

the teacher, as some might believe, but rather should come from a variety of places, such as

testing a model using technology like Desmos or consulting a colleague in the class. As long

as teachers are facilitating opportunities for students to receive meaningful feedback, then

students can continue to progress towards the goals of the PBL.

Additionally, according to the NCTM (2014) teaching practices, effective teachers

should be supporting productive struggle, which can only be done well if teachers are

tracking student progress and know when their students are struggling unproductively.

NCTM (2014) recommends that teachers elicit and use evidence of student thinking and pose

purposeful questions to formatively assess student progress towards the learning goals. They

also should facilitate meaningful mathematical discourse (NCTM, 2014), which can be done

using the strategies outlined by Smith and Stein (2011, Table 3.1). Mathematical discourse is

a great form of feedback for students and should be used regularly to assess student

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understanding and provide students with the opportunity to engage with each other’s

reasoning (Herbel-Eisenmann, et. al., 2013). Meaningful mathematical discourse and strong

critique and revision, especially peer critique, can only occur in a safe environment built to

support failure and iterative improvement (Fancher & Norfar, 2019). Therefore, teachers

must build and maintain the culture (PBLWorks, 2019) that allows for the PBL experiences

to occur by using the teaching practices discussed.

Supporting students to give and receive feedback (Grossman, et. al., 2019) is another

place where math teachers might feel a little more uncomfortable. There are fears that

students will give the wrong feedback to their peers, which Langer-Osuna (2017) points out

certainly does occur sometimes, such as when students who dominate socially are believed

without question. However, Langer-Osuna (2017) encourages teachers to continue to foster

meaningful mathematical discourse by intentionally positioning students to author ideas,

which Smith and Stein (2011) contend can be achieved through deliberate selecting and

sequencing. Discourse can also happen when students engage with each other’s ideas and

reasoning (Herbel-Eisenmann, et. al., 2013). Thus, according to Fancher and Norfar (2019),

math teachers must provide a safe environment in which failure occurs and “build the culture

[by] explicitly and implicitly promoting student independence and growth, open-ended

inquiry, team spirit, and attention to quality” (PBLWorks, 2019, p.1). This will allow many

recommendations to be accomplished: students will engage with each other’s thinking

(NCTM, 2014), be curious about that thinking (Fancher & Norfar, 2019), and provide and

receive feedback in a productive way (PBLWorks, 2019).

The last part of the iterative process is to support students to reflect and revise

(Grossman, et. al., 2019). The revision piece of this subcategory has been discussed at length,

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but the reflection piece has been largely ignored until now. Reflection is one of the

PBLWorks (2019) design elements because it is central to student growth. Through

reflection, students critique their own work and improve the next iteration of the model,

which is a process that mathematicians must endure in order to find the optimal model

(NCTM, 2014). According to PBLWorks (2019), this reflection also allows teachers to

“identify when [their students] need skill-building, redirection, encouragement, and

celebration” (p.1). This helps students realize that learning is a cyclical process of

production, feedback, reflection, and revision (Grossman, et. al., 2019). It is important that

students know how to reflect so that, when they receive feedback from testing their products,

as well as from their teachers and fellow students, they are able to strategize about how to

revise their product for improvement in the next iteration. Jansen (2020) wrote in detail about

this process, calling it rough-draft math, which is a term intended to remind teachers that, just

like in the process of writing, the learning process begins with ideas that are not completely

refined but become refined with time, reflection, and collaboration. She argues that

conceptual understanding and procedural fluency come from the opportunity for students to

be “imperfect but precise, unfinished and unsure” (Jansen, 2020, p.350) in their thinking, and

then allowed to reflect, “revisit and revise” (p. 351) their thoughts as they learn more and

gain a better understanding. Thus, this process also supports NCTM’s (2014) mathematical

teaching practice of “building procedural fluency from conceptual understanding” (p.10). As

students revise and test their mathematical models, they continue to practice the procedures

involved, which allows them to build the procedural fluency. If they start from a place of

conceptual understanding, then their reflections can be more efficient in developing

improvements on their models.

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Core Practice 4: Collaboration

The fourth and final core practice in Grossman, et. al.’s (2019) framework is

collaboration. They touch on two aspects of this: “support students to collaborate and support

students to make choices” (Grossman, et. al., 2019, p.47). Supporting students to collaborate

is very similar to the PBLWorks (2019) practice of supporting students to give and receive

feedback, so the connected frameworks to this subcategory have been discussed in detail.

Something to keep in mind when supporting students to collaborate is that students need

tools for “justifying whether particular mathematical ideas are reasonable or correct and press

one another for explanations” (Langer-Osuna, 2017, p. 239). This can be modeled when

facilitating meaningful mathematical discourse (NCTM, 2014) in small groups and whole

class discussions, and by implementing tasks (NCTM, 2014) that have been intentionally

selected to “require a range of strengths and strategies to solve” (Langer-Osuna, 2017, p.

240). Teachers have also utilized group roles and scripted protocols that support students

facilitating their own collaborative mathematical discourse (NSRF, 2021).

Supporting students to make choices is also strongly tied to the PBLWorks

framework. In fact, they have a student voice and choice (PBLWorks, 2019) category in their

design elements, where students are supported to make decisions throughout the project.

Notably, students and teachers are collaborating at every step of the process. In the

PBLWorks (2019) “Design and Plan” stage, teachers are designing for some degree of

student voice and choice, and sometimes are even designing and planning the project with

students. As they manage activities with students, they work with students to organize tasks

and schedules that will work well for both parties, with a few teacher-set deadlines to help

guide the decisions. Fancher and Norfar (2019) have their students keep track of what they

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know and need to know, and design most of their tasks around what their students suggest

they need. Additionally, they require students to identify how they are going to learn each

‘need-to-know,’ and only a small percentage of those resources are allowed to be ‘from the

teacher’ (Fancher & Norfar, 2019). This collaborative learning process also helps students

build procedural fluency from conceptual understanding and support productive struggle

(NCTM, 2014) because “teachers and students share in the intellectual authority of the work”

(Langer-Osuna, 2017, p. 239). Effective teaching practices place student thinking at the

center of the learning, and teachers work alongside students to help them understand how

their ideas connect to the mathematical learning goals of the day.

After analyzing these frameworks, it became clear that there are very strong

connections between effective project-based teaching practices and effective mathematical

teaching practices. Although some mathematical teaching practices are not as obviously

apparent upon first read, they are all essential in serving different components of the PBL

experience, and can be used to make sense of how to utilize the PBL frameworks in the

mathematics classroom.

Framework for PBL in the Mathematics Classroom

The connections between all of the frameworks for effective teaching practices in

math and PBL revealed a need for a single framework that fully captures everything

discussed above. Thus, based on previously mentioned research this paper presents a

framework (Figure 3.4) for implementing PBL in the mathematics classroom, which was

developed by merging the core connections discussed in the previous section. The framework

consists of three phases: planning, working, and sharing.

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Figure 3.4

Teaching Practices for PBL in the Mathematics Classroom (see the Appendix for larger view

of the model)

Planning Phase

Align to Standards. The first thing teachers should do is establish mathematical

goals and an authentic context that are aligned to course standards. It is important to start

with the standards because a project will only be able to drive the learning experience if it is

designed to support the goals of the standards (Condliffe, et. al., 2016). Fancher and Norfar

(2019) point out that some projects are best when inspired by a standard and then connected

to an authentic or relevant topic, while others are more interesting when inspired by a topic

and then connected to a standard. Either way, the project must be designed to create an

authentic learning experience in which it makes sense for students to need to learn the

mathematics in the established mathematical goals. Otherwise, Condliffe, et. al. (2016) warn,

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the project runs the risk of become tangential to the learning, possibly interesting and

motivating, but not purposeful in the grand scheme of the learning goals.

For example, if a teacher needs to teach standards related to exponential functions,

they might plan to investigate bacteria growth from drinking contaminated water and the

potential effects. They will determine what learning goals need to be met, such as students

need to be able to write an exponential function from a situation, graph an exponential

function from an equation, and explain the key features of an exponential function. Now that

the mathematical goals have been identified and the authentic context has been determined,

the teacher is ready to move to the next part of the framework.

Working Phase

Next, the project enters an iterative phase, in which three main components are

happening simultaneously: engage in disciplinary practices, build procedural fluency, and

assess student progress. This phase is the most important phase because this is when the

student learning occurs. As all teachers know, even the best laid plans can fall apart in the

hands of students, so this phase requires that teachers remain in tune with their students.

Engaging students in disciplinary practices. At this stage, teachers have a choice as

to how much planning they need to do before they introduce students to the project. They

could choose to prepare in detail how they will engage students in disciplinary practices or

choose to involve the students in collaboratively planning with them as the project

progresses. Either way, this stage begins with exploring a problem that needs solving,

collaborating both with colleagues and teacher, and then building, testing, and revising a

mathematical model of the problem in order to find a solution, just as one would do as a

practicing mathematician. This should be an iterative process, with students regularly testing

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their theories as well as seeking feedback from the teacher, external experts, and peer

colleagues. The students are active participants in the way that the project moves forward.

Their thinking should drive the conversations that occur in the classroom, and purposeful

questioning should be used to elicit curiosity that pushes student thinking towards the

mathematical goals (Fancher & Norfar, 2019). There should be many iterations of the model,

so Fancher and Norfar (2019) state that teachers must support students as they test, fail, learn

from the test, and continue to improve the model. They should also be encouraged, both by

the teacher and the nature of the problem, to continue to improve the model, even if it works

the first time. Take note, this is the first opportunity for teachers to involve an authentic

audience in the form of external experts that students are encouraged (or required) to consult.

In the bacteria example, teachers and students may start by connecting with the

school’s biology teacher and learn about different examples of bacteria so they can

investigate how those different types of bacteria grow. They may find some datasets that

measure bacteria growth over time and do a regression on that data using a technology tool

like Desmos to make sense of the growth. Maybe they have the opportunity to partner with

the science class and collect some of their own bacteria data, or establish contact with a

disease expert at a local lab who studies bacteria growth and can share some of their

expertise with the students. If none of that is possible, then the teacher might provide a

realistic, albeit fictional, dataset on bacteria growth for students to explore. Then, they will

start modeling the bacteria growth: taking a look at the graphical representation and learning

how to describe it, fitting an equation to the dataset, learning about the various components

of the equation and what they represent, and then thinking about what all of that means to

someone trying to stop bacteria from killing children. They may learn about vaccines and

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how those behave, possibly in the opposite way as an exponential decay model, and think

about how vaccines could be distributed to countries that still don’t have them. It is possible

that the teacher will even find in the process of collaborating with their students that other

mathematical standards can be addressed as well. Then they will communicate their findings

to an audience, having to practice describing the graph and what it means, as well as

important aspects that the audience should understand, which means they will reach the goal

of understanding key features of an exponential graph. This is just one way that a project like

this could go, since it is in large part up to the students, the teacher, and the people in the

community that they are able to engage in this effort.

Build Procedural Fluency. During this process, teachers should help students build

procedural fluency by eliciting student reasoning from conceptual understanding, personal

connections, and iterations of the mathematical model. Procedural fluency is developed from

practice of concepts that have been deeply understood and grounded in experience. This

experience may come from prior mathematical knowledge or non-mathematical personal

experiences, but it is essential that teachers are working to make these connections between

new information and prior experiences explicit for students. Otherwise, it will be difficult to

achieve fluency in the sense that the procedure comes naturally and can be applied flexibly.

Teachers can support their fluency development by eliciting student reasoning as they make

conceptual and personal connections (NCTM, 2014). In addition to these explicit supports,

students will be able to practice related procedures as they revise their mathematical model,

since each time they revisit the process in order to improve their model. Students should

begin to recognize that certain mathematical procedures are more efficient than others in

certain situations, and develop flexibility with applying the most efficient or effective

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procedure for a particular context (Knuth, 2000). As they test and improve their model, as

well as critique their peers’ models, they should experience the practice that they need to

become procedurally fluent. Teachers can, of course, incorporate tasks that support this

practice if they notice shortcomings in certain areas from the natural progression of the

project.

To continue with the bacteria example, students could find themselves fitting an

exponential model to a set of data, in which the teacher might ask how their model would

change if they found out that some of the records in the dataset were bad and needed to be

thrown out. Or they may learn that vaccines work as exponential decay, killing off bacteria at

the opposite exponential rate from when they grew, so the student might investigate how

long a child could live with bacteria growing in them and how quickly a vaccine could kill

the bacteria. This situation would have them creating multiple models with different starting

values of bacteria, and might lead them to wonder about timing and how that affects

distribution. In this project, different groups of students would most likely be studying

different types of bacteria, so the teacher could create practice for her students through a peer

review cycle, where they have to set up a model themselves before being able to give valid

feedback to their peers. This experience would give students lots of procedural practice,

while still serving the needs of the project.

Assess Student Progress. Throughout the project, teachers should also be assessing

student learning using formative assessment such as mathematical discourse, purposeful

questions, peer feedback, personal reflection, and a variety of learning tasks. Mathematical

discourse allows students to author ideas and engage with each other’s ideas, which often

leads to adjustments in design and improvement upon previous notions. Teachers can foster

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this discourse using a variety of strategies, such as Herbel-Eisenmann, et. al.’s (2013) six

teacher discourse moves: “waiting, inviting student participation, revoicing, asking students

to revoice, probing a student’s thinking, and creating opportunities to engage with another’s

reasoning” (p.182). The three that specifically support assessment of student understanding

are asking students to revoice, probing a student’s thinking, and creating opportunities to

engage with another’s reasoning. When teachers use these moves, they can see if other

students in the room are able to make sense of what their colleague has presented, summarize

it in a meaningful way, or explain the thought in more depth. Additionally, it supports the

teacher’s understanding of student learning across the room as well as things that still need to

be addressed.

Purposeful questioning helps teachers make sense of student reasoning as well as to

gather evidence of student learning. It can be used in whole class discussions, but is often

most effective in small group or individual settings where students feel safe to author ideas

that could be incorrect (Langer-Osuna, 2017). According to NCTM (2014), questioning

should be open-ended, with the goal of understanding student reasoning, rather than

imparting what the teacher believes the student should be saying through a series of

questions.

Peer feedback serves multiple purposes: giving students more disciplinary

experiences, giving students the opportunity to engage with each other’s reasoning and

practice skills, learning about what is missing in their own mathematical model, and showing

the teacher how well students can analyze work and provide accurate and helpful feedback.

The process of giving feedback requires students to think about and understand what their

peers did in order to assist them. The act of receiving critiques from peers requires the ability

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to reflect and decide what is beneficial and what should be ignored. Teachers should help

students reflect on all of this feedback, as the reflection process will also benefit students in

improving their models in addition to showing teachers how well students are understanding

the mathematics.

Learning tasks are also forms of formative assessment, and provide structure to

students as they explore the mathematics. This is where math teachers will feel most

comfortable, because they have been utilizing learning tasks to teach for a long time. The key

in PBL is to pace them in a way that supports student progress without giving anything away

to students before it is necessary, in contrast to the traditional pace of providing students with

the information that teachers know they will need before they experience any reason to use it.

A strategically placed learning task will help teachers to be able to focus student learning on

a mathematical goal, and limit students from spending valuable time learning ideas that do

not support the learning targets of the project. The timing is important, however, because

teaching mathematics before there is a need for it could diminish the learning that happens

during the project. Teachers should have a way to monitor student progress on the learning

task, such as written responses or utilizing the teacher dashboard on a technology tool such as

Desmos; otherwise they will have a hard time assessing what students have learned and what

they still need to learn.

In the bacteria example, students might start with an exploration of a data set

representing bacteria growth, letting technology fit an exponential model to the graph for

them, and making observations about that graph and equation. After sparking curiosity

around exponential models, the teacher could then teach a lesson on exponential equations

and what each part represents. This can be done either by examining the different equations

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from each group’s model and identifying patterns that students notice between their contexts

and their models, or by teaching a general model with a few basic examples and having

students return to their models and analyze them with their new knowledge. Then, with the

purpose of learning how to communicate with an expert about the models, the teacher could

do a lesson on key features of an exponential graph, having students practice describing

various examples to each other. Many learning tasks during a PBL unit will look very similar

to ones that are done in a non-PBL classroom, with the main difference being that teachers in

a PBL unit should be continuously connecting learning tasks back to the challenging

question, and it should be clear to students how the learning task will serve their needs in the

project.

Assessment of student learning should be happening throughout the learning process,

in every conversation, every task, and every opportunity to engage with the students.

Traditional assessments are also appropriate to sprinkle in throughout the PBL experience,

but they should not be the only form of learning evidence collected (Fancher & Norfar,

2019). Teachers will want to include at least one summative assessment during the process as

well, and it is encouraged to do so by creating a rubric for the final product, as well as for

periodic check-in’s through the course of the project. For example, since the learning goals

were to be able to write, graph and describe an exponential function, the rubric would need to

assess whether they were able to do that. Additionally, teachers should provide intentional

opportunities for students to reflect on their work, since a lot of learning can occur when

people reflect on their own progress towards a goal (PBLWorks, 2019).

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Sharing Phase

Find Authentic Audience. The final piece of the puzzle is to find an authentic

audience that can engage in the students’ solutions. This could be integrated into multiple

phases of the project, but should at least be at the end of the project. When there is an

authentic audience, or a public product as PBLWorks (2019) calls it, producing quality work

becomes much more important to students. Just to reiterate, teachers can find many authentic

audiences in their communities, such as university experts, parents of students with specific

expertise, school leaders, or local policymakers. They can use these adults to serve as clients

or consultants in realistic situations that the teacher created (Fancher & Norfar, 2019), unless

they can find adults who have an actual need from the students, which is even better. If

teachers cannot find adults in related fields to serve in this role, Fancher and Norfar (2019)

suggest to still bring in external adults to role play for students. Fortunately, people are often

excited to be able to engage with students about their expertise, so many times all it takes is

for a teacher or student to reach out and ask. Additionally, remember to look for

opportunities for students to be able to make a contribution to their community or the world

as this can make the work even more powerful to them (Himes, et. al., 2020).

For the bacteria example, students could consult with bacterial disease experts at the

local hospital or university, learning about how bacteria grow and how vaccines work to

combat that growth. For social action, they could find a group, such as Doctors Without

Borders, that works to distribute vaccines to areas where bacteria growth causes diarrheal

deaths, and raise money or awareness about that work. They could write children’s books

teaching about how bacteria grow exponentially, and what effects that has on children around

the world, and then go to the local elementary school to read their books. In another idea,

51

they could create an infographic that informs policymakers of their discoveries, and then

meet with policymakers and advocate for them to help distribute lifesaving vaccines to these

areas. Teachers who come up with lots of exciting options like these might decide to provide

students with the opportunity to choose, which is an alternate way to fulfill the PBLWorks

(2019) recommendation of student voice and choice.

The Teaching Practices for PBL in the Math Classroom framework was developed by

merging best practices frameworks in both PBL and mathematics to become a resource for

math teachers looking to implement PBL in their classes. It is made up of three phases:

planning, working, and sharing. In the planning phase, the most important thing for teachers

to do is to identify the mathematical learning goals and an associated authentic context. The

rest of the planning can either happen during this phase or the working phase. In the working

phase, there are three main components that teachers should be doing simultaneously and

iteratively: engaging students in disciplinary practices, building procedural fluency, and

assessing student progress. Then, the final phase is the sharing phase, where teachers need to

find an authentic audience for their students, or support their students in finding their own

authentic audience. This framework was designed to be able to support math teachers who

are just beginning with PBL as well as to be a resource for math teachers to consult once they

have become comfortable teaching with PBL.

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CHAPTER 4: CONCLUSION

This chapter concludes the paper with a discussion of the limitations related to the

literature review and the recommendations for future studies that can assist in addressing

those limitations. With so much work left to do in order to understand the effects of having a

math PBL framework as a resource, the author is excited to see what future research comes

from these recommendations.

Limitations and Recommendations

There is still a lot to be learned about how to effectively implement PBL in a

mathematics classroom, and how to know if PBL is an effective learning tool for the

mathematics classroom. Condliffe, et. al. (2016) points out that there have only been a few

empirical studies done on the effectiveness of PBL for teaching math, which they say is most

likely due to the relatively few math teachers attempting to use PBL. Thus, this paper has had

to rely on studies done in other disciplines, such as science and English, as well as studies

done on effective mathematics practices that, as previously discussed, are very similar to

PBL practices. The results of the studies that have been done on PBL in math classes should

be accepted with caution, since it is possible that the students would have outperformed their

peers regardless of the method used (PBL versus traditional) because the schools were not

randomly selected (Condliffe, et. al., 2016). Similarly, there are very few studies that look at

how to prepare math teachers to be able teach using PBL, the effectiveness of the programs

that are preparing teachers to use PBL, how to support them through the process once they

begin, or the effectiveness of the coaching that is being done to support them. There have

been some studies on the effectiveness of teacher preparation for PBL, such as Himes, et.

al.’s (2019) assessment of how well teachers applied their learning after being prepared to

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utilize the PBI Global model and Germuth and EvalWorks’ (2018) evaluation of teacher

improvement after participating in the SummerSTEM PBL teacher preparation program,

which both saw improved teaching practices and have had math teachers participate.

However, these studies have not focused specifically on math teachers, and anecdotally, it

was found in the PBI Global that actually the math teachers needed a lot more support and

were barely incorporating the project into their practice (David Hardt, personal

communication, October 16, 2020).

Analysis of the PBL frameworks and math teaching practices suggests that PBL could

be effective, and the framework proposed in this paper is designed to provide teachers,

coaches, researchers, and administrators with a tool that could be used to plan, implement

and assess a strong PBL unit in a math class. The next step for this framework would be to

assess its validity, utilizing an evaluation rubric that is designed to reveal evidence of the

framework in a classroom. This rubric should be developed and then used to test the

comprehensiveness and usability of the framework for understanding how to implement PBL

in the mathematics classroom. Once the framework has been validated, it should be used to

develop professional development for math teachers on how to utilize it, since one of the

barriers is teacher preparation (DiBiase & McDonald, 2015). The framework should also be

used in partnership with math teachers to design and implement PBL in a mathematics

classroom. This would allow researchers to analyze how effective it is in supporting math

teachers as they implement PBL, and further support professional development for its use.

Conclusion

With a brief look at education history, it is evident that PBL is a tried and true

practice and that it provides teachers with structures that can help them create the learning

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environments recommended by math education experts. Despite this evidence, math teachers

find it challenging to implement PBL and there are very few math-specific PBL resources for

interested teachers to learn from. The studies that have been done show that students who

learn via PBL demonstrate more growth in their learning than their traditional counterparts.

Additionally, students and teachers who use PBL tend to be more motivated to work hard and

produce quality work. Thus, it is important that more math-specific PBL resources be

developed and more math teachers overcome the barriers to embracing PBL as a teaching

method. This paper is a start to that effort, providing a framework that merges best practices

in math education with best practices in PBL. The framework is based on three phases,

planning, working, and sharing, which should be used as a reference guide for teachers as

they design projects and as they reflect on how well the project went. Hopefully, the

framework and examples in this paper help math teachers feel confident that they can

successfully implement PBL in their classrooms.

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APPENDIX

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Figure 3.4

Teaching Practices for PBL in the Mathematics Classroom

Planning Assess Student Progress: Use mathematical discourse, purposeful

questions, peer feedback, and a variety of learning tasks.

Working

Engage Students in Disciplinary Practices:

Develop tasks that support students as they collaboratively explore a problem that needs

solving, then build, test and revise a mathematical model to find a solution.

Build Procedural Fluency: Elicit student learning from conceptual

understanding, personal connections to the work, and iterations of the mathematical

model.

Find Authentic Audience: Create an opportunity for students

to share their solutions with someone beyond their class,

preferably making a contribution to their community in some way.

Sharing

Align to Standards: Establish math goals and

authentic context.